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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > meaiuninc3 | Structured version Visualization version GIF version |
Description: Measures are continuous from below: if 𝐸 is a sequence of nondecreasing measurable sets (with bounded measure) then the measure of the union is the limit of the measures. This is the general case of Proposition 112C (e) of [Fremlin1] p. 16 . This theorem generalizes meaiuninc 46437 and meaiuninc2 46438 where the sequence is required to be bounded. (Contributed by Glauco Siliprandi, 13-Feb-2022.) |
Ref | Expression |
---|---|
meaiuninc3.p | ⊢ Ⅎ𝑛𝜑 |
meaiuninc3.f | ⊢ Ⅎ𝑛𝐸 |
meaiuninc3.m | ⊢ (𝜑 → 𝑀 ∈ Meas) |
meaiuninc3.n | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
meaiuninc3.z | ⊢ 𝑍 = (ℤ≥‘𝑁) |
meaiuninc3.e | ⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀) |
meaiuninc3.i | ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1))) |
meaiuninc3.s | ⊢ 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) |
Ref | Expression |
---|---|
meaiuninc3 | ⊢ (𝜑 → 𝑆~~>*(𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | meaiuninc3.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ Meas) | |
2 | meaiuninc3.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
3 | meaiuninc3.z | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑁) | |
4 | meaiuninc3.e | . . 3 ⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀) | |
5 | meaiuninc3.p | . . . . . 6 ⊢ Ⅎ𝑛𝜑 | |
6 | nfv 1912 | . . . . . 6 ⊢ Ⅎ𝑛 𝑘 ∈ 𝑍 | |
7 | 5, 6 | nfan 1897 | . . . . 5 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑘 ∈ 𝑍) |
8 | meaiuninc3.f | . . . . . . 7 ⊢ Ⅎ𝑛𝐸 | |
9 | nfcv 2903 | . . . . . . 7 ⊢ Ⅎ𝑛𝑘 | |
10 | 8, 9 | nffv 6917 | . . . . . 6 ⊢ Ⅎ𝑛(𝐸‘𝑘) |
11 | nfcv 2903 | . . . . . . 7 ⊢ Ⅎ𝑛(𝑘 + 1) | |
12 | 8, 11 | nffv 6917 | . . . . . 6 ⊢ Ⅎ𝑛(𝐸‘(𝑘 + 1)) |
13 | 10, 12 | nfss 3988 | . . . . 5 ⊢ Ⅎ𝑛(𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1)) |
14 | 7, 13 | nfim 1894 | . . . 4 ⊢ Ⅎ𝑛((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1))) |
15 | eleq1w 2822 | . . . . . 6 ⊢ (𝑛 = 𝑘 → (𝑛 ∈ 𝑍 ↔ 𝑘 ∈ 𝑍)) | |
16 | 15 | anbi2d 630 | . . . . 5 ⊢ (𝑛 = 𝑘 → ((𝜑 ∧ 𝑛 ∈ 𝑍) ↔ (𝜑 ∧ 𝑘 ∈ 𝑍))) |
17 | fveq2 6907 | . . . . . 6 ⊢ (𝑛 = 𝑘 → (𝐸‘𝑛) = (𝐸‘𝑘)) | |
18 | fvoveq1 7454 | . . . . . 6 ⊢ (𝑛 = 𝑘 → (𝐸‘(𝑛 + 1)) = (𝐸‘(𝑘 + 1))) | |
19 | 17, 18 | sseq12d 4029 | . . . . 5 ⊢ (𝑛 = 𝑘 → ((𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1)) ↔ (𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1)))) |
20 | 16, 19 | imbi12d 344 | . . . 4 ⊢ (𝑛 = 𝑘 → (((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1))) ↔ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1))))) |
21 | meaiuninc3.i | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1))) | |
22 | 14, 20, 21 | chvarfv 2238 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1))) |
23 | meaiuninc3.s | . . . 4 ⊢ 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) | |
24 | nfcv 2903 | . . . . . 6 ⊢ Ⅎ𝑘𝑀 | |
25 | nfcv 2903 | . . . . . 6 ⊢ Ⅎ𝑘(𝐸‘𝑛) | |
26 | 24, 25 | nffv 6917 | . . . . 5 ⊢ Ⅎ𝑘(𝑀‘(𝐸‘𝑛)) |
27 | nfcv 2903 | . . . . . 6 ⊢ Ⅎ𝑛𝑀 | |
28 | 27, 10 | nffv 6917 | . . . . 5 ⊢ Ⅎ𝑛(𝑀‘(𝐸‘𝑘)) |
29 | 2fveq3 6912 | . . . . 5 ⊢ (𝑛 = 𝑘 → (𝑀‘(𝐸‘𝑛)) = (𝑀‘(𝐸‘𝑘))) | |
30 | 26, 28, 29 | cbvmpt 5259 | . . . 4 ⊢ (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) = (𝑘 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑘))) |
31 | 23, 30 | eqtri 2763 | . . 3 ⊢ 𝑆 = (𝑘 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑘))) |
32 | 1, 2, 3, 4, 22, 31 | meaiuninc3v 46440 | . 2 ⊢ (𝜑 → 𝑆~~>*(𝑀‘∪ 𝑘 ∈ 𝑍 (𝐸‘𝑘))) |
33 | fveq2 6907 | . . . 4 ⊢ (𝑘 = 𝑛 → (𝐸‘𝑘) = (𝐸‘𝑛)) | |
34 | 10, 25, 33 | cbviun 5041 | . . 3 ⊢ ∪ 𝑘 ∈ 𝑍 (𝐸‘𝑘) = ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) |
35 | 34 | fveq2i 6910 | . 2 ⊢ (𝑀‘∪ 𝑘 ∈ 𝑍 (𝐸‘𝑘)) = (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) |
36 | 32, 35 | breqtrdi 5189 | 1 ⊢ (𝜑 → 𝑆~~>*(𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 Ⅎwnf 1780 ∈ wcel 2106 Ⅎwnfc 2888 ⊆ wss 3963 ∪ ciun 4996 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5689 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 1c1 11154 + caddc 11156 ℤcz 12611 ℤ≥cuz 12876 ~~>*clsxlim 45774 Meascmea 46405 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-disj 5116 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-omul 8510 df-er 8744 df-map 8867 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-acn 9980 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-rlim 15522 df-sum 15720 df-struct 17181 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-mulr 17312 df-starv 17313 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-rest 17469 df-topn 17470 df-topgen 17490 df-ordt 17548 df-ps 18624 df-tsr 18625 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-lm 23253 df-xms 24346 df-ms 24347 df-xlim 45775 df-salg 46265 df-sumge0 46319 df-mea 46406 |
This theorem is referenced by: (None) |
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